# Complexity of Game Dynamics

### Alex Fabrikant

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/EECS-2008-113

September 9, 2008

### http://www.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-113.pdf

What happens when independent agents interact based on their selfish interests? Game theory's earliest and most natural simple model of such interaction is the best-response Nash dynamics, the process of agents making unilateral moves that are their best response to the actions of others. Pure Nash equilibria, when they do exist, arise naturally as the steady states of this process. When they don't exist, behavioral predictions can be made from "sink equilibria", a universal (guaranteed to exist) generalization of Nash equilibria to "steady clusters of states."

We now know that it is vital for a model of naturally-occuring behavior to be computationally tractable, not only for simulations, but also to check how realistic the model is, since we expect that nature (aside maybe from quantum physics) cannot produce systems with fundamentally more computational power than computers as we know them.

In this dissertation, I show several results about the tractability of analyzing a game's best-response dynamics. If the game payoff tables are given explicitly (in normal form), searching for pure Nash or sink equilibria is trivial. However, most interesting games are represented more succinctly. For pure Nash equilibria, I show that in potential games, a well-studied class of succinct games guaranteed to have pure equilibria, finding pure Nash equilibria is PLS-complete, and the best-response dynamics takes an exponentially long time to converge in the worst case, even when the games are restricted to network routing games, or to symmetric games. For sink equilbria, I prove that it is PSPACE-complete to analyze them in graphical games.

On the practical side, I resolve a decade-old well-known open problem in networking by establishing that it is PSPACE-complete to predict whether Internet inter-domain routing may be destabilized by large-scale oscillations; that is, whether a system of path preferences in the BGP protocol may lead to flapping. This turns out to be a question about the best-response dynamics in a special kind of game.

Lastly, I propose several enhanced equilibrium concepts inspired by game dynamics that allow for higher rationality by the players while mostly retaining the tractability and universality of sink equilibria in normal-form games.

**Advisor:** Christos Papadimitriou

BibTeX citation:

@phdthesis{Fabrikant:EECS-2008-113, Author = {Fabrikant, Alex}, Title = {Complexity of Game Dynamics}, School = {EECS Department, University of California, Berkeley}, Year = {2008}, Month = {Sep}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-113.html}, Number = {UCB/EECS-2008-113}, Abstract = {What happens when independent agents interact based on their selfish interests? Game theory's earliest and most natural simple model of such interaction is the best-response Nash dynamics, the process of agents making unilateral moves that are their best response to the actions of others. Pure Nash equilibria, when they do exist, arise naturally as the steady states of this process. When they don't exist, behavioral predictions can be made from "sink equilibria", a universal (guaranteed to exist) generalization of Nash equilibria to "steady clusters of states." We now know that it is vital for a model of naturally-occuring behavior to be computationally tractable, not only for simulations, but also to check how realistic the model is, since we expect that nature (aside maybe from quantum physics) cannot produce systems with fundamentally more computational power than computers as we know them. In this dissertation, I show several results about the tractability of analyzing a game's best-response dynamics. If the game payoff tables are given explicitly (in normal form), searching for pure Nash or sink equilibria is trivial. However, most interesting games are represented more succinctly. For pure Nash equilibria, I show that in potential games, a well-studied class of succinct games guaranteed to have pure equilibria, finding pure Nash equilibria is PLS-complete, and the best-response dynamics takes an exponentially long time to converge in the worst case, even when the games are restricted to network routing games, or to symmetric games. For sink equilbria, I prove that it is PSPACE-complete to analyze them in graphical games. On the practical side, I resolve a decade-old well-known open problem in networking by establishing that it is PSPACE-complete to predict whether Internet inter-domain routing may be destabilized by large-scale oscillations; that is, whether a system of path preferences in the BGP protocol may lead to flapping. This turns out to be a question about the best-response dynamics in a special kind of game. Lastly, I propose several enhanced equilibrium concepts inspired by game dynamics that allow for higher rationality by the players while mostly retaining the tractability and universality of sink equilibria in normal-form games.} }

EndNote citation:

%0 Thesis %A Fabrikant, Alex %T Complexity of Game Dynamics %I EECS Department, University of California, Berkeley %D 2008 %8 September 9 %@ UCB/EECS-2008-113 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-113.html %F Fabrikant:EECS-2008-113